Chapter 5 gives the convected wave equation for a small velocity-potential…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow
Signal Speeds in Fluids, Ch Steady compressible flow: energy, stagnation and the Mach number
Problem
Chapter 5 gives the convected wave equation for a small velocity-potential perturbation ϕ′ in a uniform stream U along x, with free-stream sound speed c∞: (∂t∂+U∂x∂)2ϕ′=c∞2(∂x2∂2ϕ′+∂y2∂2ϕ′). For steady flow past a thin body ϕ′ does not depend on t.
((a)) Show that the steady form of this equation is (1−M∞2)ϕxx+ϕyy=0,M∞=U/c∞.
((b)) For M∞>1 this is a wave equation in which x plays the part that time plays in Chapter 2's shallow-water characteristics, where a disturbance travels along dx/dt=u±c. Write the equation in the form ϕxx=β−2ϕyy with β=M∞2−1, find its two families of characteristic lines in the (x,y) plane, and identify them with the Mach lines, giving the Mach angle μ in terms of M∞.
((c)) Explain, from the sign of 1−M∞2, why running the same construction at M∞<1 gives no real characteristic directions, and relate this to the absence of any confined zone of silence in the subsonic flow that instead obeys the Prandtl-Glauert rule.
((d)) A thin wedge in a stream at M∞=1.8 turns the flow through θ=3.0∘ at a sharp corner, taken as the origin, with the flow along +x and y measured away from the wall. Upstream of the corner the static pressure is p∞=80.0kPa and γ=1.4. State whether the point (x,y)=(10.0m,4.0m) lies inside the region reached by the corner's disturbance, and, if it does, use Ackeret's formula Cp=2θ/β together with the ideal-gas relation ρU2=γpM2 to find the static pressure there.